Let me ask also, why do you think your son would get an A-/B+ on the pre-calc exam, if he took it now? Trig is widely used calculus, and I can’t see how anyone could do all of the AP Calc AB questions in a book without having a pretty good command of trig.
I don’t think there’s a point in pushing for 100% on the pre-calc test. People occasionally make a stupid mistake or two and it doesn’t mean that they need to consolidate their understanding.
Thanks everyone. This was very helpful and thought provoking. Seems like consensus is between #2 and #5 and we will mull these two over the next few weeks and have our son take some preclac and calc AB exams as well to calibrate.
UCB Alumnus, Thank you very much for the exams. I had my son take the precalc one.
https://math.berkeley.edu/sites/default/files/pages/F03_Final_Exam-B.Johnson.pdf
He couldn’t do 9 and 11 but could do the bonus problem so scored 94. I am OK with him not getting 9 as he doesn’t know what simple harmonic motion is and once I explained it to him he got that too. However since he didn’t get 9 I feel that his base in trig is not solid.
QuantMech, it is not the silly mistake, it’s the inability to intuitively understand trig identities that worries me. Using trig is easy and enough for Calc BC but that doesn’t mean one has got the concept of trig. I belong to the Russian school of thinking when it comes to math which is that repeated practice is what makes perfect aka drill and kill. Or as Malcolm Gladwell would say - 10,000 hours of practice is needed to truly master something.
MV Calc/Linear algebra will be taught in school and not in a college. It wouldn’t include much differential equations but only high level discussions. So that he will have to do on his own. I should factor that in as I don’t think he should do real analysis without doing differential equations. I am not sure where he will take his college course as I haven’t looked into that yet. I am also not thinking about college admissions for now but just what math courses he should take to have a decent foundation in math. But from your post as well as that of others I realize that there are more variables to consider.
SlackerMomMD, The math professor is suggeting option #5 as Sylvan, Tollfree and Deadpirit have suggested as well.
DJCordiero, Thanks. A long time back when he was a baby I taught him set theory using EPGY materials which brings back a lot of fond memories. (If you move two apples from the fruits set to the vegetables set, how many fruits remain in the fruits set?) I have access to the courses it is the sequence that is worrying me and the Stanford sequence is less accelerated than what he wants.
I’m not anti-Stats at all. Statistics is fascinating and your son might like it, but AP Stats is a waste of time for him. Instead of that, either he should wait to take a real, calc-based statistics class at a college as a junior or senior, or he should wait until college and take a real, calc-based statistics class then.
If he is not strong in selected bits of the precalculus course, it may be a waste of time for him to spend a while year taking the course just to cover the bits he does not know. It may be worth having him focus his self-study and practice this summer on those bits (instead of the entire course/book), then continue on to the next course.
I’m with Tollfree, Sylvan, Deadspirit: sequence 5 is the best. He’d be bored to death in precalc. He got a 94 without actually taking the class - it means that in a couple weeks he’ll have that content down pat. It’d be a complete waste of time and probably incredibly frustrating for him (kind of like forcing a kid who’s easily running bicycle races with kids twice his age to stay on a tricycle because he fell once). It actually sounds painful to me to force him into precal (and I started on this thread ready to recommend 2, too. But your child is just too fast and too advanced for Precalc Honors at a HS pace.)
If he’s interested in stats, have him take calculus-based stats for fun at a local college (summer course?)
LateCut, sorry to keep asking for clarification, but I took a look at the Berkeley math test and at problem 11 in particular. When you say that your son couldn’t get it, I assume you mean that he got the right hand side to 2 sin(theta) cos(theta), but then could not prove that equal to sin(2 theta)?
I think as a fundamental you cant have too strong a calculus background. This often includes a class in Dif Eq as well as MV Secondly linear algebra is used everwhere. I think that statistics and CS are very fundamental to everyday living and I would take them over an Algebra or Analysis class. For some reason many math people seem to shy away from statistics classes. Yet statistics and probability are so important in everything in life. There are college statistics classes that don’t use calculus such as ANOVA or linear regression Before many kids take Algebra or Analysis they take classes on how to do proofs. That can be very important. There are all sorts of variations of algebra and analysis classes. Is the class an introduction to algebra or do they do Galois theory by the end of it . What book does the Analysis class use? Lots of people think Complex analysis is much more important than real analysis. I think getting a strong foundation in the fundamentals and some of the practical aspects of math is more important than rushing to an algebra and analysis class that your son might just have to retake in college
The 4th one is a little funky… Presumably he does like math right? He should probably take AP stats concurrently with Calc BC and take Real & Complex Analysis junior year. Once he has taken Real & Complex Analysis he will know more about where he wants to go with math so he can then make a further decision on what to take next, there’s not much of a natural extension to them as there is up until it.
Agree that statistics knowledge is useful – but high school AP statistics is likely to be a bore for such a strong math student. Better for the student to take calculus-based probability and statistics courses at a college.
I have known strong HS math students that did not find AP stats boring Martin Hairer recently won the fields medal and much of his work is in probability and statistics fields. I just think that statistics becomes too much of an after thought to too many math students
At many high schools, AP statistics is where weaker-in-math students are steered to in order to take an “AP” math course that is not as hard as calculus. Note that AP statistics is not calculus-based, so a strong math student who knows calculus may find a calculus-based statistics course to be more interesting and useful. Since taking college math courses while in high school is under consideration, taking calculus-based college statistics instead of high school AP statistics can be considered.
most college statistics at the undergraduate level are not calculus based. Probability classes can be calculus based Calculus based statistics is usually reserved for graduate level statistics. You may want to double check. Undergraduate statistics currently involve lots of R and SAS programming but generally not much if any calculus. There is also lots of other interesting math that doesnt include calculus( combinatorics, set theory etc)
While non-calculus-based statistics may be the most common at the introductory level (which AP statistics is supposed to be like), statistics majors are generally required to take calculus-based statistics courses.
For example, the University of Florida’s statistics major has these requirements:
http://www.stat.ufl.edu/academics/ugrad/undergrad_req.shtml
Note that STA 4321 Introduction to Probability requires MAC 2313 Calculus III.
Students who just want a one course introduction to statistics at the University of Florida can take STA 3032 Engineering Statistics which is calculus-based instead of the non-calculus-based STA 2023 Introduction to Statistics (which AP statistics is considered a substitute for).
http://www.stat.ufl.edu/academics/ugrad/undergrad_ulcourses.html
@florida26 Maybe the students who didn’t find AP Stats boring had a class like the one taught at our school. They covered the AP Stats in the first semester and then did real calculus based stats the second semester with the teacher teaching calculus as he went along. No one is saying that stats and probability aren’t hugely important, my dh, a biolgist, complains all the time that his students don’t know enough stats. The problem is that the AP Stats curriculum is very watered down.
I agree with mathmom. At my university, there are several levels of introductory statistics, and the “good” level requires multi-variable calculus as a pre-requisite. AP statistics is likely to be a real let-down after Calc BC. I would not recommend that sequence. I recommend that the OP take a look at the released AP Statistics exams. I anticipate that he will be startled to discover that students must state the obvious in order to get full points on some of the free response questions. For example, if you have a statistical test that requires at least 5 cases per “box,” and the minimum number of cases in any of the “boxes” is 12, you have to write down that 12 > 5 or you lose a point. My spouse calls this “compliance mathematics.”
There are some useful things in an AP Stats course. However, I think that a good math student could teach him/herself all of it in 6 weeks, squeezed in among normal responsibilities of a full-time class load + extra-curriculars (to say nothing of the summer).
The reason that I raised the question earlier about problem 9 on the Berkeley exam [whether you son got to 2 sin (theta) cos(theta) and then could not prove that it was equal to sin (2 theta)?] is this: I don’t think that the students were expected to prove that step on the Berkeley exam, but just to recognize the identity. I base that on looking at the level of the rest of the questions and having a pretty good idea of how pre-calculus math tends to be taught in the U.S., even at Berkeley. There is plenty of practice with double-angle and half-angle formulas in Calc BC.
Is there any advantage at his school for him to just take the AP CS test some May? Any CS electives for which it is a prerequisite or dual-enrollment classes? My son took the AP CS test when he was in 8th grade. All he really needed was to read some of the review book to learn what sorts of things would be on the test and what parts of Java actually are not included (and so can’t be used in the answers). A practice test let him get the feel for writing code on paper (ick). Taking that test early has been helpful in his applications to do summer research things and he is eligible to take CS classes at whatever level he thinks would be good at the community college.
I agree that it sounds like he would be bored in Precalc. My son skipped Precalc, and now as a sophomore in Calc BC finally feels like the math is at the appropriate level and pace for him. He did have to study the trig identities a bit on his own. I think the content of Precalc varies at different schools. Here it also has some vectors, matrices, various coordinate systems, some sums & series, and a bit of functions leading into calc. He’d worked with vectors in physics and most of the other topics at a math circle and through self-teaching for programming projects. Not all of these topics are used in Calc I, so maybe just find out what topics are taught in Precalc at his school – since that may be different from the placement tests into Calc.
QuantMech, He took one look at problem #11 and gave up without trying. Last night over dinner I asked him to do the problem in his head and guided him through. I asked him to reduce the RHS to just sine and cosine and he easily got to 2sin(θ)cos(θ) and then said that’s how far he can go. Then I asked him to expand the LHS and he said he doesn’t know how to do that but when my wife asked him to remember what he studied last summer about sin(α + β) he easily got it. I am disappointed because these are things he should just know without even having to think about it. The algebraic manipulation he can do in his head but he needs to remember and completely absorb all the log and trig identities which I believe only comes from the Russian method of drill and kill. He can figure out the geometric solution to sin(α + β) quite easily and demonstrated that to us last night but he can’t remember the identity. He has a big problem with anything that requires memorization and it is frustrating really.
However I agree with UCBalumnus that it doesn’t make sense to repeat a whole class just to pick up a month’s worth of materials. So at this point we are leaning towards option #5 but will have him repeat log and trig problems over and over till he can do them without any thinking. The log identities he typically derives on the fly and then uses them and I want him to intuitively just know it.
Ynotgo, he may take AP CS and AP Stats on his own. We usually are quite laissez-faire when it comes to his education and he will have to decide that for himself. I think he can study both on his own over 9th grade and take the exams at the end of 9th grade. Both are fairly easy courses. But I don’t know if he will want to do that.
I thought #5 was also most appropriate or #14. AP stats would likely be boring. In college as a math major, we could only take the graduate level Stat course (calculus-based) to fulfill any major requirements. I can’t imagine a kid who is in love with rings, fields and groups being engaged in AP Stats. I’ve also sat through this type of class in graduate school (for social science professional degree) - it’s a snoozer.
The key for young students is to engage and challenge them. I always advocate a solid base. If the OP is really worried about his precalc base, then maybe an refresher class over the summer?
From an American standpoint, if your son can do the derivations, I would say he is totally okay. In the U.S. system, there is not a lot to be gained from the type of fluency with trig identities that you seem to be concerned about. In my experience, when students are working with them, they remember them. Then the identifies may lie dormant for a while. When they are needed, they are quickly relearned or recalled. I will send you a message with my suggestions about the math sequence.
What is the rush? This kid is only in 8th grade and would get a B in pre calc per op yet we already have him programmed to take real analysis. If he would get a B in pre calc how does he know any calculus at all? It is one thing to know a little bit about groups rings and fields and another thing to read Langs book on Algebra and understand it. It is also one thing to know a little bit about python and Java and another thing to know how to make extensive web sites, apps and solve problems using extensive data bases in the cloud. I would take it one step at a time